The muon identification system of the ALICE experiment at the CERN LHC is based on Resistive Plate Chamber (RPC) detectors. These RPCs are operated in the so-called maxi-avalanche mode with a gas mixture made of tetrafluoroethane (C$_{2}$H$_{2}$F$_{4}$), sulfur hexafluoride (SF$_{6}$) and isobutane (i-C$_{4}$H$_{10}$). All of these components are greenhouse gases: in particular, the first two gases are already phasing out of production, due to recent European Union regulations, and their cost is progressively increasing. Therefore, finding a new eco-friendly gas mixture has become extremely important in order to reduce the impact of the RPC operation on the environment, and for economic reasons. Due to the similar chemical structure, hydrofluoroolefins appear appropriate candidates to replace C$_{2}$H$_{2}$F$_{4}$ thanks to their very low GWPs, especially tetrafluoropropene (C$_{3}$H$_{2}$F$_{4}$) with the trade name HFO1234ze. In order to identify an eco-friendly gas mixture fulfilling the requirements for operation in the ALICE environment in the coming years, a dedicated experimental set-up has been built to carry out R&D studies on promising gas mixtures. Measurements have been performed with a small-size RPC equipped with the front-end electronics, providing signal amplification, developed for ALICE operation at high luminosity after the LHC Long Shutdown 2. HFO1234ze-based mixtures with the addition of CO$_{2}$ are discussed in this paper as well as the role of i-C$_{4}$H$_{10}$ and SF$_{6}$ as quenchers in such mixtures.
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